Let ABCD be a quadrilateral. Suppose AB and CD have equal length and angles BAD and BCD are supplementary (i.e., angle BAD plus angle BCD equals 180 degrees). Show that AD is parallel to BC.

One easy solution I can think of is to glide reflect quadrilateral ABCD so that A' is where C is, and B' is where D is. Because BAD=B'A'D'=DCD' and BAD+ BCD equal 180, then D'CD + BCD = 180, so line BC=line BD' and by a similar proof we can prove line AD equals line AC', and since BAD+B'C'D'=180, then BA and D'C are parallel and equal, which is a condition for ABD'C' being parallelogram, and so the sides (AD and BC) are parallel.

B---------CA'------D'
  \         / /            \   
=A------DB'---------C'

Posted by Gamer on 2005-05-28 17:09:21